To a casual observer, I imagine that the minor firestorm in various math blogs that my columns generated might suggest that my remarks injected something new to the mathematics education field. But, in fact, everything I said has been written about and discussed in the mathematics education community for some forty years or so, and essentially (though not in every detail) agreed upon. Discussed and agreed upon by people who have spent their professional careers studying those issues and have formed carefully thought out conclusions that have been subjected to, and passed, professional peer review.

This month's column presents some of the data on which I based by original postings. That, of course, makes the column much longer than usual. On the other hand, I'll be quoting from some of the leading mathematics education scholars of the twentieth and twenty-first centuries, so I hope you think it's worth it.

K-8 mathematics education is perhaps more complicated than other subjects because there are two major domains involved: mathematics and mathematics education. My area of expertise is the former. For much of my career, I knew little of the latter, and was, quite frankly, somewhat dismissive of it. Then, as a result of various career moves, I found myself interacting more and more with members of the math ed community, culminating in my appointment to the Mathematical Sciences Education Board a few years ago. And that experience opened my eyes to just how ignorant I was of the problems inherent in mathematics education, and how we professional mathematicians, while clearly having something important to contribute to mathematics education, are manifestly unable to do it alone. Mathematical cognition is simply too complicated a subject to handle properly without multiple sources of expertise. (As is usually the case, perhaps the most valuable outcome of learning, definitely true in my case, is a better appreciation of the limits of what we know, and where our ignorance begins. I think Richard Feynman once said something along those lines. If he didn't, he should have.)

Piaget said it first

The complexities inherent in learning multiplication, and the importance of noting the distinction between multiplication and addition, were first pointed out (to the best of my knowledge) in Piaget's groundbreaking work in the 1960s and 70s. (See, for example, Jean Piaget, The Child's Conception of Number, Norton, 1965, or Piaget, Grize, Szeminska and Bangh, Epistomology and the Psychology of Functions, Reidel, 1977.) Piaget presented evidence-based arguments to show that multiplication is fundamentally different from addition, and should be taught as such. In various writings, he suggested ways multiplication could be taught, and presented evidence to show that children as young as six were able to grasp some of the basic ideas that underlie multiplication.

More recently, mathematics education experts Terezina Nunes (London University) and Peter Bryant (Oxford University) devote two entire chapters (Chapters 7 and 8) of their excellent 1996 book Children Doing Mathematics to the distinction between multiplication and repeated addition, explaining in some detail the many subtle problems inherent in mastering multiplication.

Arguably Nunes and Bryant know more about multiplication and how to teach it than any other researchers in the world, though it would appear that language translation difficulties prevent their work from reaching many US-based bloggers. (The Brits, of course, spell "maths" with a plural "s" and often use an "s" where we know it should be a "z".)

The language translation problem clearly causes further problems for many
US-based commentators, since, in addition to the excellent and exhaustive work of Nunes and Bryant, a lot of the other research into the nature of multiplication, how people perceive it, and how best to teach it, has also been carried out in the United Kingdom. (Perhaps this is because British education researchers are unhampered by the Math Wars that continue to rage in the US, or by the often unhealthy influence of large textbook publishers we work under, or by the influence of lay local education boards that so often plagues US teachers.)

A further impediment to acquainting oneself with the research that has been done into the teaching and learning of multiplication - and the consequences of doing it wrongly - seems to be the difficulty of typing the term "repeated addition" into a well known search engine, which in a recent brief experiment I conducted brought up most of the references I cite below - generally on the first page of returns.

Okay, I'm teasing. Teasing and provoking are part of the stock-in-trade of columnists who think an issue is sufficiently important to be given an airing.
And I do think the "multiplication is not repeated addition" issue really is important. (Moreover, the scope of the problem is widespread. According to some studies I'll cite later, the belief that the two are the same is close to universal.) That's why I am breaking the columnist's rule of never going back to the same theme more than once. But, hey, the same rule holds for movies, and the new Batman did really well this summer. 'Tis the season.

Why do I think this particular issue is important? Why do I insist we should avoid teaching young children that multiplication is repeated addition? Well, one very good reason is that it is just plain wrong. That ought to be enough of a reason, particularly in mathematics, which is all about precision and correctness, but for many people - brought up to believe multiplication actually is repeated addition - that does not seem to be sufficient. So I'll have to take a sledgehammer approach and bring in some of the wealth of evidence to support my claim. (Actually, I have a better reason. Some teachers and several homeschooling parents emailed me and asked for further information.)

At this point, newcomers to this little mini-drama should read my two previous columns on the matter. (Something few of the bloggers who "responded" actually did, since most of the comments I saw before I gave up looking at them were about things I neither claimed nor implied, and in many cases had stated the exact opposite to. Maybe I still inadvertently use British linguistic constructs that baffle some readers.) Everything that follows presumes you have read the two earlier episodes, and for the most part I won't repeat here points I made there. Now read on ...

The bloggers' arguments

By and large, the bloggers' arguments boiled down to six kinds:

1. We've always taught it that way so why should we change?

2. I learned it that way and it worked for me. (This claim was almost invariably followed by statements that indicated that the writer had actually come out of the learning process with some serious misconceptions, proving my point not theirs.)

3. Repeated addition might be wrong from a mathematician's perspective, but it works just fine for positive whole numbers and most of us don't need arithmetic beyond positive whole numbers.

4. It might be technically wrong, but we can correct it later, and what harm does it do in the long run?

5. There's no other way to do it.

6. There really isn't a problem. Teachers don't teach multiplication as repeated addition; rather they already do what I am calling for, so I was attacking a straw man.

There were also contributions having little to do with my argument. For instance, a number of people (maybe the same person, since blog contributors seem to post by pseudonyms) confused the fact that multiplication on the natural numbers can be reduced to addition on the natural numbers with whether or not addition and multiplication are the same arithmetic operations. (They are not.) I mentioned the former reduction procedure in one of my original articles, but reducibility of one mathematical abstraction to another tells us little about the relationship between the cognitive operations of concern. It doesn't even tell us they are the same formal mathematical operation. Reducibility - which is not the same as equality - is of interest to pure mathematicians and philosophers; connections and differences between the operations students learn, understand, and perform are what matters in K-8 math education. There was also confusion about the significance of the standard definition of equality for Dirichlet-style functions, another red herring in the present context.

Of the six argument types listed above, only the last three arguments merit a response. (Note that 5 and 6 are contradictory.) In fact, I gave responses to 4 and 5 in my original articles, so rather than repeat here what I wrote already, I'll re-address them here (along with point 6) by letting the words of the math ed experts speak for me.

[By the way, I sympathize with those respondents who noted that, while I might have identified a genuine and serious problem, I had not given a specific prescription of what should be done to correct it. Indeed I did not, for the reason that I don't presume to be an expert in how to teach mathematics at the K-8 level. My perspective is of someone who has to teach the students who find their way to university, and find that some of them do not understand the basic operations of arithmetic. Of course, I've read much of the work of the math ed experts, so I know the results they've obtained. But that hardly makes me an expert in that field. This is a column in an e-zine of a professional mathematical society, with an editor and an advisory board, and I can be fired at any moment, as would surely happen were I to pretend to be something I am not.]

Multiplication is a tricky concept

Here is what Nunes and Bryant have to say about multiplication at the start of Chapter 7 of the book I cited above. (There are more British spellings in their work, so the bloggers won't be able to read this valuable resource, though doing some research before launching into an Internet tirade is clearly not their forte.) The previous chapter deals with addition and subtraction.

According to [a common view] there need be no major change in children's reasoning [after they have mastered addition and subtraction] in order for them to learn how and when to carry out multiplication and division. This view was challenged by Piaget and his colleagues [...] who suggested that understanding multiplication and division represents a significant qualitative change in children's thinking.

There certainly are significant discontinuities between addition and subtraction on the one hand and multiplication and division on the other... but there are some significant continuities too... the continuities and discontinuities are as important as each other, and both need to be thoroughly charted if we are to understand the many steps that every child has to take towards a full understanding of multiplication.

After acknowledging that there remains some controversy surrounding multiplication, particularly how you classify types of multiplication, the authors continue - and this is really important:

We must begin with a word of caution. Multiplicative reasoning is a complicated topic because it takes different forms and it deals with many different situations, and that means that the empirical research on this topic is complicated too. So, in order to make sense of the empirical work, we must first spend some time setting up a conceptual framework for the analysis of children's reasoning and only then go on to review the research. [...] Up to now it was possible to build the concepts and the vocabulary needed slowly through the chapter; with multiplication and division we stray so far from common sense and everyday vocabulary that we have to agree on a set of terms and conceptual distinctions at the outset.

It's important to note that the particular difficulty Nunes and Bryant are alerting the reader to here is that required in understanding how children learn and do multiplication. The situation facing the child is a different one, namely learning what multiplication is, how to do it, and when and how to use it. But the task facing the researchers is tricky precisely because the concept is itself tricky.

[I suspect that most of us who end up being classified as "good at math" simply learn how to multiply, in a mechanical fashion, and only later come to understand what it is, if we reach such understanding at all. An interesting feature of those blog threads I mentioned is how they indicate that a fair number of people never reach the understanding stage.]

The authors' above caution notwithstanding, I would strongly urge anyone teaching children to do arithmetic should read and reflect on (at least) Chapters 7 and 8 of the Nunes-Bryant book. Although not a "how to teach" book - it is a report of research findings - it does provide a wealth of ideas for how to do it. In particular, the authors indicate why teaching multiplication should involve exposing the students to three distinct kinds of problem situation: one-to-many correspondence situations (pp.143-146), situations involving co-variation between variables (pp.146-149), and situations that involve sharing and successive splits (pp.149-153).

Nunes and Bryant's work alone, I think, takes care of question 5 above.

Incidentally, here is what Nunes and Bryant have to say about repeated addition (p.153):

The common-sense view that multiplication is nothing but repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

They go on to enumerate and describe some of the more salient complexities of multiplication. The issue is far too complex for me to summarize effectively here. It takes Nunes and Bryant an entire chapter and then some. But note what they are saying in the above quoted passage: Even in the special case of the positive whole numbers, where repeated addition gives the answer to a multiplication sum, the two are not at all the same.

The problem with brittle metaphors

What about question 4? Why not start out by teaching children that multiplication is repeated addition, and then, when they meet cases where it is not (negative numbers, fractions, irrational numbers - hey, isn't that actually most of the numbers?), simply get them to change their concept. Again, I provided a number of answers in my previous articles.

One is that the metaphor or model we first use to grasp a new concept is invariably the one that continues to dominate long after we have theoretically "learned" that it was not correct. Math ed specialist Ann Watson of Oxford University makes the same point in her article School mathematics as a special kind of mathematics (watch out for some more confusing British spellings) [http://www.math.auckland.ac.nz/mathwiki/images/4/41/WATSON.doc.]:

Additive to multiplicative reasoning: A shift from seeing additively to seeing multiplicatively is expected to take place during late primary or early secondary school. Not everyone makes this shift successfully, and multiplication seen as 'repeated addition' lingers as a dominant image for many students. This is unhelpful for learners who need to work with ratio, to express algebraic relationships, to understand polynomials, to recognise and use transformations and similarity, and in many other mathematical and other contexts.

Mike Askew and Margaret Brown, in their (British-English) paper How do we teach children to be numerate?
[http://www.bera.ac.uk/publications/pdfs/520668_Num.pdf]
make the same point (page 10):

... research has shown that multiplication as repeated addition and division as sharing appear to be widely understood by primary aged children. However, ... understanding the meaning of multiplication is more complex (Nunes and Bryant, 1996) and difficulties with fully understanding multiplication and division persist into secondary school (Hart, 1981).

There is evidence that such early ideas - multiplication as repeated addition and division as sharing - have an enduring effect and can limit children's later understandings of these operations. For example, understanding multiplication only as repeated addition may lead to misconceptions such as 'multiplication makes bigger' and 'division makes smaller' (Hart 1981, Greer 1988). Even with older children researchers have shown that they may persist with using primitive methods such as repeated addition or repeated subtraction with larger numbers (Anghileri, 1999).

For more of the same, but this time with American spelling, there is Thompson and Saldanha's article Fractions and Multiplicative Reasoning, in Kilpatrick, Martin, and Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics, pp. 95-113, published by the National Council of Teachers of Mathematics, 2003. They say (page 103):

[...] multiplication is not the same as repeated addition. [...] One may engage in repeated addition to evaluate the result of multiplying, but envisioning adding some amount repeatedly cannot support conceptualizations of multiplication. [...] Generally, most students do not see proportionality in multiplication.

The authors go on to acknowledge (lament?) that a lot of instructors continue to perpetuate the problem:

In fact, a large amount of curriculum and instruction has the explicit aim that students understand multiplication as a process of adding the same number repeatedly. But an extensive research literature documents how "repeated addition" conceptions become limiting and problematic for students having them (de Corte, Verschaffel, & Van Coillie, 1988; Fischbein et al., 1985; Greer, 1988b; Harel, Behr, Post, & Lesh, 1994; Luke, 1988).

You can, of course, check out any of those cited sources for yourself. Note that this addresses point 6 in my above list, but I'll come back to that later.

A controlled experiment

Want more? How about a controlled experiment? Jee-Hyun Park and Terezinha Nunes of Oxford Brookes University report in their paper The development of the concept of multiplication, published in the journal Cognitive Development, Volume 16, Issue 3, July-September 2001, pages 763-773 (watch out for more of those pesky British spellings):

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. This study assessed the two hypotheses through an intervention method. It was hypothesised that an intervention based on the origin of the concept of multiplication would be more effective than the one that did not offer the learners a conceptual basis for learning. Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. Both groups made significant progress from pre- to posttest. The group taught by correspondence made significantly more progress in multiplicative reasoning than in additive reasoning problems. The group taught by repeated addition made similar progress in both types of problems. At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest. Thus, this study supports the hypothesis that the origin of the concept of multiplication is in the schema of correspondence rather than in the idea of repeated addition.

Okay, I know you are desperate for more, but this will have to be the last. (Yes, more Brits, I'm afraid.)

One more

In Teaching and Learning Primary Numeracy: Policy, Practice and Effectiveness
(A review of British research for the British Educational Research Association in conjunction with the British Society for Research in the Learning of Mathematics), Editors: Mike Askew and Margaret Brown
[http://www.bera.ac.uk/publications/pdfs/numeracyreview.pdf],
you will find, on pages 12-13:

The research on children's understanding of multiplicative reasoning has so far had less impact on teaching than the research on additive reasoning. Classifications of multiplicative reasoning problems do not have the same privileged treatment in the teaching of primary teachers, where the difficulties of multiplicative reasoning are often ignored. Research has identified the common misconception that multiplication makes bigger and division makes smaller and provided evidence that this misconception is likely to be connected to the concept of multiplication as repeated addition and division as repeated subtraction. Nevertheless, teachers continue to be encouraged to use these very ideas in teaching.

Progress in the investigation of multiplicative reasoning includes the following:

children's understanding of commutativity of multiplication is a later development than commutativity of addition and is also influenced by problem type (Nunes & Bryant, 1996);

children's understanding of distributivity is also a late development (Nunes & Bryant, 1996);

children in infant classes already show some basic knowledge of multiplication and division (Bryant, Morgado, & Nunes, 1992); Nunes et al., 1993) with the understanding of the inverse relation between the divisor and the quotient in division lagging behind the ability to solve sums with the support of manipulative materials (Bryant et al., 1992);

children are able to use their understanding of multiplication to solve division questions much earlier than they are able to think of using division strategies to solve multiplication problems (Nunes et al.,1993);

children's understanding of inverse relations when considering multiplicative relationships appears much later than expected by some mathematics educators in the past little contextual variation in performance in multiplicative reasoning tasks has been found but significant effects of the mathematical terminology used were documented (Nunes et al., 1993).

... Recommendations to teach multiplication as repeated addition and division as repeated subtraction are also cause for concern. Research suggests that such teaching may be at the root of later misconceptions. Alternative models for teaching have been shown more effective in experimental studies (Clark & Nunes, 1998) but evidence is still limited and more research is urgently needed.

The problem is widespread

There's much more along the same lines, folks. Yet, despite forty years worth of research findings and reports mostly (but not always) saying the same thing, the word has still not spread beyond the mathematics education community. Heavens, few professional mathematicians are familiar with this research. (I admitted already that I was not until relatively late in my career.)

Indeed, not long ago, researcher Ann Dowker of Oxford University (yes, the one the UK) asked 38 educated adults to define whole number multiplication, and apart from a few who gave a vacuous answer such as "to multiply," all defined it as repeated addition. (Individual Differences in Arithmetic, Psychology Press, 2005, page 43.)

In Canada (remember they tend to use British spelling as well), Brent Davis carried out a similar study recently and got similar results. (See Brent Davis and Dennis J. Sumara, Complexity And Education: Inquiries Into Learning, Teaching, And Research, Lawrence Erlbaum Associates, 2006.)

The above citations all address the bloggers' argument 6. Now, I am confident that a lot of teachers make a great job of teaching basic arithmetic well. There is, as I have tried to indicate here, no shortage of good educational resources on the matter (along with a lot of badly written material). But the research reported in some of the articles I just cited suggests that not all do. In fact, what the empirical studies show is that the belief that multiplication is repeated addition is prevalent, indeed dominant. Though I had noticed this false belief (and the analogous one that exponentiation is repeated multiplication) in some incoming university students in my classes over many years - where it causes significant problems in calculus and other subjects - I never really recognized that there is a serious problem here until I attended a presentation Brent Davis made on his survey a year or so ago. In a world dominated by important issues that are multiplicative or exponential, for leaders and citizens to have an additive-grounded quantitative sense is dangerous in the extreme.

The mathematician's reason

All of the reasons mentioned above for not teaching multiplication as repeated addition focus on the problems children have as they make their way up the educational ladder, constantly having to unlearn one definition and relearn another. (Many of those bloggers I mentioned at the start clearly never recovered from the misconceptions they acquired when they first met multiplication.) As I mentioned before, my perspective on this issue is having to deal with university students who have not acquired an understanding of basic arithmetic adequate to get them through their degree courses.

By the time students graduate from high school, they should have a good conceptual and procedural understanding of basic arithmetic on the integers, the rationals, and the reals, and they should know that it is the same arithmetical operations on all three domains. This expectation, at least, ought to be well known and uncontroversial. It is stated clearly, and up front, in what is generally regarded as the "Bible" of K-8 mathematics education in the US, namely the book Adding It Up: Helping Children Learn Mathematics, authored by the Mathematics Learning Study Committee of the National Research Council, and published by the National Academies Press in 2001. (It's a great resource that every math teacher and every homeschooling parent should read and consult regularly.)

On page 72, you will find the following:

Number Systems

At first, school arithmetic is mostly concerned with the whole numbers: 0, 1, 2, 3, and so on. The child's focus is on counting and on calculating- adding and subtracting, multiplying and dividing. Later, other numbers are introduced: negative numbers and rational numbers (fractions and mixed numbers, including finite decimals). Children expend considerable effort learning to calculate with these less intuitive kinds of numbers. Another theme in school mathematics is measurement, which forms a bridge between number and geometry.

Mathematicians like to take a bird's-eye view of the process of developing an understanding of number. Rather than take numbers a pair at a time and worry in detail about the mechanics of adding them or multiplying them, they like to think about whole classes of numbers at once and about the properties of addition (or of multiplication) as a way of combining pairs of numbers in the class. This view leads to the idea of a number system. A number system is a collection of numbers, together with some operations (which, for purposes of this discussion, will always be addition and multiplication), that combine pairs of numbers in the collection to make other numbers in the same collection. The main number systems of arithmetic are (a) the whole numbers, (b) the integers (i.e., the positive whole numbers, their negative counterparts, and zero), and (c) the rational numbers-positive and negative ratios of whole numbers, except for those ratios of a whole number and zero.

Thinking in terms of number systems helps one clarify the basic ideas involved in arithmetic. This approach was an important mathematical discovery in the late nineteenth and early twentieth centuries. Some ideas of arithmetic are fairly subtle and cause problems for students, so it is useful to have a viewpoint from which the connections between ideas can be surveyed.

Of course, Adding It Up is aimed at K-8 education, so the report goes only as far as arithmetic on the rationals. But the authors want teachers to prepare the way for the students to progress all the way through to the real numbers. A short while later, on page 94, they caution:

The number systems that have emerged over the centuries can be seen as being built on one another, with each new system subsuming an old one. This remarkable consistency helps unify arithmetic. In school, however, each number system is introduced with distinct symbolic notations: negation signs, fractions, decimal points, radical signs, and so on. These multiple representations can obscure the fact that the numbers used in grades pre-K through 8 all reside in a very coherent and unified mathematical structure - the number line.

Which brings us back to where I came in with my two original columns on the subject: the need for mathematics education at any stage to (1) reflect the significant changes that have taken place in mathematics over the past hundred-and-fifty years; (2) be consistent with, and prepare the student for a (possible) progression to, mathematics as it is actually practiced in today's world; and (3) take account of the masses of scholarly research that has been carried out in mathematics education in the past fifty years.

And that's pretty well all there is to it, really.

The blogs

Of the blogs I looked at, which had threads devoted to my "repeated addition" columns, the following all had some good, thoughtful comments by their owners
and by some of their contributors - by no means all agreeing with me - though the discussion in "Let's Play Math" soon descended to uninformed and repetitive name calling, and the owner eventually closed the thread, which unfortunately soon reappeared elsewhere.

Those blogs each have interesting and useful things to say on other math ed topics as well. Most of the others I saw seem to be little more than sounding boards for people who are so convinced that the overwhelming mass of evidence must all be wrong - since it runs counter to their beliefs - they don't even bother to read it. Those (other) blogs are not information exchanges from which you can learn anything, but platforms for people to espouse their own particular, unsubstantiated and often wildly wrong beliefs. Mathematicians who care about our subject and who like to think that the students who pass through our classrooms emerge with a good understanding of the mathematics we taught them, should be advised that they venture into any other mathematics education blog at their own risk.

Meanwhile, now you know why (or at least you know where to start finding out why) it is crucially important that we not teach children that multiplication is repeated addition. (Or, if you prefer, why we should not teach them in a way that leaves them believing this!)

Closing thought

The entire episode sparked by my original columns led me to reflect on a rather curious and unfortunate state of affairs regarding mathematics education that I shall end with.

While most of us would acknowledge that, while we may fly in airplanes, we are not qualified to pilot one, and while we occasionally seek medical treatment, we would not feel confident diagnosing and treating a sick patient, many people, from politicians to business leaders, and now to bloggers, feel they know best when it comes to providing education to our young, based on nothing more than their having themselves been the recipient of an education. How did our society ever reach the stage where some of us are so willing to ignore the painstaking work of professionals who have spent their lifetime studying education? Being the recipient of some service is generally an important prerequisite to becoming a provider of that service, but it usually requires a lot of learning, training, study, and practice to become and to be a provider of that service. That holds for education as much as for flying airplanes or treating the sick.